two zeros of cubic polynomial ax^3+3x^2-bx-6 are -1 and -2. find the third zero and values of a and b

Answer 1

if x=-1, then -a + 3 + b - 6 = 0
a - b = -3
if x=-2, then -8a + 12 + 2b - 6 = 0
8a - 2b = 6
4a - b = 3
subtract them:
3a = 6
a = 2
in a-b=-3
2-b = -3
b = 5

so expression is
2x^3 + 3x^2 - 5x - 6

which is (x+1)(x+2)( ?x + ?)
by common sense, the last bracket must be
(2x -3)

so the third root is x=3/2

check:
(x+1)(x+2)(2x-3)
= (x^2 + 3x + 2)(2x-3)
= 2x^3 - 3x^2 + 6x^2 - 9x + 4x - 6
= 2x^3 + 3x^2 - 5x - 6 , all is good

Answer 2

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